Quadratic Functions

Roots of Quadratic Equations and the Quadratic Formula

In this section, we will learn how to find the root(s) of a quadratic equation. Roots are also called x-intercepts or zeros. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one, two, or zero roots.

When we are asked to solve a quadratic equation, we are really being asked to find the roots. We have already seen that completing the square is a useful method to solve quadratic equations. This method can be used to derive the quadratic formula, which is used to solve quadratic equations. In fact, the roots of the function,

f (x) = ax2 + bx + c

are given by the quadratic formula. The roots of a function are the x-intercepts. By definition, the y-coordinate of points lying on the x-axis is zero. Therefore, to find the roots of a quadratic function, we set f (x) = 0, and solve the equation,

ax2 + bx + c = 0.

We can do this by completing the square as,

Solving for x and simplifying we have,

Thus, the roots of a quadratic function are given by,

This formula is called the quadratic formula, and its derivation is included so that you can see where it comes from. We call the term b2 −4ac the discriminant. The discriminant is important because it tells you how many roots a quadratic function has. Specifically, if

1. b2 −4ac < 0 There are no real roots.

2. b2 −4ac = 0 There is one real root.

3. b2 −4ac > 0 There are two real roots.

We will examine each case individually.

Case 1: No Real Roots

If the discriminant of a quadratic function is less than zero, that function has no real roots, and the parabola it represents does not intersect the x-axis. Since the quadratic formula requires taking the square root of the discriminant, a negative discriminant creates a problem because the square root of a negative number is not defined over the real line. An example of a quadratic function with no real roots is given by,

f(x) = x2 − 3x + 4.

Notice that the discriminant of f(x) is negative,

b2 −4ac = (−3)2− 4 · 1 · 4 = 9 − 16 = −7.

This function is graphically represented by a parabola that opens upward whose vertex lies above the x-axis. Thus, the graph can never intersect the x-axis and has no roots, as shown below,

Case 2: One Real Root

If the discriminant of a quadratic function is equal to zero, that function has exactly one real root and crosses the x-axis at a single point. To see this, we set b2 −4ac = 0 in the quadratic formula to get,

Notice that is the x-coordinate of the vertex of a parabola. Thus, a parabola has exactly one real root when the vertex of the parabola lies right on the x-axis. The simplest example of a quadratic function that has only one real root is,

y = x2,

where the real root is x = 0.

Another example of a quadratic function with one real root is given by,

f(x) = −4x2 + 12x − 9.

Notice that the discriminant of f(x) is zero,

b2 −4ac = (12)2− 4 · −4 · −9 = 144 − 144 = 0.

This function is graphically represented by a parabola that opens downward and has vertex (3/2, 0), lying on the x-axis. Thus, the graph intersects the x-axis at exactly one point (i.e. has one root) as shown below,

Case 3: Two Real Roots

If the discriminant of a quadratic function is greater than zero, that function has two real roots (x-intercepts). Taking the square root of a positive real number is well defined, and the two roots are given by,

An example of a quadratic function with two real roots is given by,

f(x) = 2x2− 11x + 5.

Notice that the discriminant of f(x) is greater than zero,

b2− 4ac = (−11)2− 4 · 2 · 5 = 121 − 40 = 81.

This function is graphically represented by a parabola that opens upward whose vertex lies below the x-axis. Thus, the graph must intersect the x-axis in two places (i.e. has two roots) as shown below,

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In the next section we will use the quadratic formula to solve quadratic equations.